cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A326783 BII-numbers of uniform set-systems.

Original entry on oeis.org

0, 1, 2, 3, 4, 8, 9, 10, 11, 16, 20, 32, 36, 48, 52, 64, 128, 129, 130, 131, 136, 137, 138, 139, 256, 260, 272, 276, 288, 292, 304, 308, 512, 516, 528, 532, 544, 548, 560, 564, 768, 772, 784, 788, 800, 804, 816, 820, 1024, 1088, 2048, 2052, 2064, 2068, 2080
Offset: 1

Views

Author

Gus Wiseman, Jul 25 2019

Keywords

Comments

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. A set-system is uniform if all edges have the same size.
Alternatively, these are numbers whose binary indices all have the same binary weight, where the binary weight of a nonnegative integer is the numbers of 1's in its binary digits.

Examples

			The sequence of all uniform set-systems together with their BII-numbers begins:
    0: {}
    1: {{1}}
    2: {{2}}
    3: {{1},{2}}
    4: {{1,2}}
    8: {{3}}
    9: {{1},{3}}
   10: {{2},{3}}
   11: {{1},{2},{3}}
   16: {{1,3}}
   20: {{1,2},{1,3}}
   32: {{2,3}}
   36: {{1,2},{2,3}}
   48: {{1,3},{2,3}}
   52: {{1,2},{1,3},{2,3}}
   64: {{1,2,3}}
  128: {{4}}
  129: {{1},{4}}
  130: {{2},{4}}
  131: {{1},{2},{4}}

Crossrefs

Cf. A000120, A029931, A048793, A070939, A047966, A306017, A306021, A319269, A320324, A326031, A326784 (regular), A326785 (uniform regular), A326788.

Programs

  • Mathematica
    bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[0,100],SameQ@@Length/@bpe/@bpe[#]&]

A326784 BII-numbers of regular set-systems.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 8, 9, 10, 11, 12, 16, 18, 25, 30, 32, 33, 42, 45, 51, 52, 63, 64, 75, 76, 82, 94, 97, 109, 115, 116, 127, 128, 129, 130, 131, 132, 136, 137, 138, 139, 140, 144, 146, 160, 161, 192, 256, 258, 264, 266, 288, 385, 390, 408, 427, 428, 434, 458
Offset: 1

Views

Author

Gus Wiseman, Jul 25 2019

Keywords

Comments

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. A set-system is regular if all vertices appear the same number of times.

Examples

			The sequence of all regular set-systems together with their BII-numbers begins:
   0: {}
   1: {{1}}
   2: {{2}}
   3: {{1},{2}}
   4: {{1,2}}
   7: {{1},{2},{1,2}}
   8: {{3}}
   9: {{1},{3}}
  10: {{2},{3}}
  11: {{1},{2},{3}}
  12: {{1,2},{3}}
  16: {{1,3}}
  18: {{2},{1,3}}
  25: {{1},{3},{1,3}}
  30: {{2},{1,2},{3},{1,3}}
  32: {{2,3}}
  33: {{1},{2,3}}
  42: {{2},{3},{2,3}}
  45: {{1},{1,2},{3},{2,3}}
  51: {{1},{2},{1,3},{2,3}}

Crossrefs

Cf. A000120, A001511, A005176, A029931, A048793, A070939, A295193, A322554, A326031, A326701, A326783 (uniform), A326785 (uniform regular).

Programs

  • Mathematica
    bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[0,100],SameQ@@Length/@Split[Sort[Join@@bpe/@bpe[#]]]&]

A327080 BII-numbers of maximal uniform set-systems (or complete hypergraphs).

Original entry on oeis.org

0, 1, 2, 3, 4, 8, 9, 10, 11, 16, 32, 52, 64, 128, 129, 130, 131, 136, 137, 138, 139, 256, 512, 772, 1024, 2048, 2320, 2592, 2868, 4096, 8192, 13376, 16384, 32768, 32769, 32770, 32771, 32776, 32777, 32778, 32779, 32896, 32897, 32898, 32899, 32904, 32905, 32906
Offset: 1

Views

Author

Gus Wiseman, Aug 20 2019

Keywords

Comments

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every set-system (finite set of finite nonempty sets) has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges.
A set-system is uniform if all edges have the same size.

Examples

			The sequence of all maximal uniform set-systems together with their BII-numbers begins:
    0: {}
    1: {{1}}
    2: {{2}}
    3: {{1},{2}}
    4: {{1,2}}
    8: {{3}}
    9: {{1},{3}}
   10: {{2},{3}}
   11: {{1},{2},{3}}
   16: {{1,3}}
   32: {{2,3}}
   52: {{1,2},{1,3},{2,3}}
   64: {{1,2,3}}
  128: {{4}}
  129: {{1},{4}}
  130: {{2},{4}}
  131: {{1},{2},{4}}
  136: {{3},{4}}
  137: {{1},{3},{4}}
  138: {{2},{3},{4}}

Crossrefs

BII-numbers of uniform set-systems are A326783.
The normal case (where the edges cover an initial interval) is A327081.
Cf. A000120, A048793, A070939, A326031, A326784, A326785, A327041.

Programs

  • Mathematica
    bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[0,100],With[{sys=bpe/@bpe[#]},#==0||SameQ@@Length/@sys&&Length[sys]==Binomial[Length[Union@@sys],Length[First[sys]]]]&]

A327081 BII-numbers of maximal uniform set-systems covering an initial interval of positive integers.

Original entry on oeis.org

1, 3, 4, 11, 52, 64, 139, 2868, 13376, 16384, 32907
Offset: 1

Views

Author

Gus Wiseman, Aug 20 2019

Keywords

Comments

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every set-system (finite set of finite nonempty sets) has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges.
A set-system is uniform if all edges have the same size.

Examples

			The sequence of all maximal uniform set-systems covering an initial interval together with their BII-numbers begins:
      0: {}
      1: {{1}}
      3: {{1},{2}}
      4: {{1,2}}
     11: {{1},{2},{3}}
     52: {{1,2},{1,3},{2,3}}
     64: {{1,2,3}}
    139: {{1},{2},{3},{4}}
   2868: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4}}
  13376: {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}
  16384: {{1,2,3,4}}
  32907: {{1},{2},{3},{4},{5}}

Crossrefs

BII-numbers of uniform set-systems are A326783.
BII-numbers of maximal uniform set-systems are A327080.
Cf. A000120, A048793, A070939, A326031, A326784, A326785, A327041.

Programs

  • Mathematica
    bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
Select[Range[1000],With[{sys=bpe/@bpe[#]},#==0||normQ[Union@@sys]&&SameQ@@Length/@sys&&Length[sys]==Binomial[Length[Union@@sys],Length[First[sys]]]]&]

A327373 BII-numbers of complete simple graphs.

Original entry on oeis.org

0, 1, 4, 52, 2868, 9112372, 141334497921844, 39614688284139543691484924724, 3138550868424102398255194438067307501961665532948002835252, 19701003098197239607207513568280927372312554341759233318802451615112823176074440555010583132712036457851366790597428
Offset: 0

Views

Author

Gus Wiseman, Sep 04 2019

Keywords

Comments

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every set-system (finite set of finite nonempty sets) has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges.

Crossrefs

BII-numbers of uniform set-systems are A326783.
BII-numbers of maximal uniform set-systems are A327080.
BII-numbers of maximal uniform normal set-systems are A327081.
Cf. A000120, A006125, A018900, A029931, A048793, A070939, A326031, A326784, A326785, A327041.

Programs

  • Mathematica
    Table[If[n==1,1,Total[2^(Total[2^#]/2&/@Subsets[Range[n],{2}])]/2],{n,0,10}]
Showing 1-5 of 5 results.

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